We construct new families of conformally invariant differential operators acting on densities. We introduce a simple, direct approach which shows that all such operators arise via this construction when the degree is bounded by the dimension. The method relies on a study of well-known transformation laws and on Weyl's theory regarding identities holding ``formally'' vs. ``by substitution''. We also illustrate how this new method can strengthen existing results in the parabolic invariant theory for conformal geometries.
Spectrum of a certain class of first order conformally invariant operators on the sphere is explicitly computed. The class contains the (elliptic verions of) Rarita-Schwinger operator and its higher spin analogues.
Conformally variational Riemannian invariants (CVIs), such as the scalar curvature, are homogeneous scalar invariants which arise as the gradient of a Riemannian functional. We establish a wide range of stability and rigidity results involving CVIs, generalizing many such results for the scalar curvature.
In this paper we provide a sharp characterization of the smooth four-dimensional sphere. The assumptions of the theorem are conformally invariant, and can be reduced to an L^2 inequality of the Weyl tensor and positivity of the Yamabe invariant.
We define an invariant for compact spin manifolds $X$ of dimension $4k$ equipped with a metric $h$ of positive Yamabe invariant on its boundary. The vanishing of this invariant is a necessary condition for the conformal class of $h$ to be the conformal infinity of a conformally compact Einstein metric on $X$.
In [5], P. Lecomte conjectured the existence of a natural and conformally invariant quantization. In [7], we gave a proof of this theorem thanks to the theory of Cartan connections. In this paper, we give an explicit formula for the natural and conformally invariant quantization of trace-free symbols thanks to the method used in [7] and to tools already used in [8] in the projective setting. This formula is extremely similar to the one giving the natural and projectively invariant quantization in [8].
We study conformal invariants that arise from functions in the nullspace of conformally covariant differential operators. The invariants include nodal sets and the topology of nodal domains of eigenfunctions in the kernel of GJMS operators. We establish that on any manifold of dimension $n\geq 3$, there exist many metrics for which our invariants are nontrivial. We discuss new applications to curvature prescription problems.
In this paper, we prove a classification theorem of 4-manifolds according to some conformal invariants, which generalizes the conformally invariant sphere theorem of Chang-Gursky-Yang \cite{CGY}. Moreover, it provides a four-dimensional analogue of the well-known classification theorem of Schoen-Yau \cite{SY2} on 3-manifolds with positive Yamabe invariants.
This paper constructs a family of conformally invariant differential operators acting on spinor densities with leading part a power of the Dirac operator. The construction applies for all powers in odd dimensions, and only for finitely many powers in even dimensions. These operators arise naturally as obstructions to formal solution of the Dirac equation on the Fefferman-Graham ambient space with prescribed boundary conditions.
In this paper we study an integral invariant which obstructs the existence on a compact complex manifold of a volume form with the determinant of its Ricci form proportional to itself, in particular obstructs the existence of a Kähler-Einstein metric, and has been studied since 1980's. We study this invariant from the view point of locally conformally Kähler geometry. We first see that we can define an integral invariant for coverings of compact complex manifolds with automorphic volume forms. This situation typically occurs for locally conformally Kähler manifolds. Secondly, we see that this …
We present a global conformal invariant on closed six-manifolds which obstructs the existence of a conformally Einstein metric. We show that this obstruction is nontrivial and, up to multiplication by a constant, is the unique such invariant. This also gives rise to a (possibly trivial) diffeomorphism invariant which obstructs the existence of an Einstein metric. We also discuss global conformal invariants which obstruct the existence of a conformally Einstein metric on closed six-manifolds with infinite fundamental group.
We show that on conformal manifolds of even dimension $n\geq 4$ there is no conformally invariant natural differential operator between density bundles with leading part a power of the Laplacian $Δ^{k}$ for $k>n/2$. This shows that a large class of invariant operators on conformally flat manifolds do not generalise to arbitrarily curved manifolds and that the theorem of Graham, Jenne, Mason and Sparling, asserting the existence of curved version of $Δ^k$ for $1\le k\le n/2$, is sharp.
We give explicit formulas for all odd order differential intertwinors on the subbundle of the bundle of spinor-$k$-forms that are annihilated by the Clifford multiplication over the odd dimensional standard sphere. The Dirac and Rarita-Schwinger operators appear in the case of $k=0$ and $k=1$, respectively.
This paper presents conformal invariants for Riemannian manifolds of dimension greater than or equal to four whose vanishing is necessary for a Riemannian manifold to be conformally related to an Einstein space. One of the invariants is a modification of the Cotton tensor, the other is a $n$--dimensional version of the Bach tensor. In general both tensors are smooth only on an open and dense subset of $M$, but this subset is invariant under conformal transformations. Moreover, we generalize the main result of "Conformal Einstein Spaces in $N$--Dimensions" published in \emph{Ann. Global Anal. G…
We discuss a general scheme for a construction of linear conformally invariant differential operators from curved Casimir operators; we then explicitly carry this out for several examples. Apart from demonstrating the efficacy of the approach via curved Casimirs, this shows that this method applies both in regular and in singular infinitesimal character, and also that it can be used to construct standard as well as non--standard operators. The examples treated include conformally invariant operators with leading term, in one case, a square of the Laplacian, and in another case, a cube of the L…
Let ${\cal F}_λ$ be the space of tensor densities on ${\bf R}^n$ of degree $λ$ (or, equivalently, of conformal densities of degree $-λn$) considered as a module over the Lie algebra $so(p+1,q+1)$. We classify $so(p+1,q+1)$-invariant bilinear differential operators from ${\cal F}_λ\otimes{\cal F}_μ$ to~${\cal F}_ν$. The classification of linear $so(p+1,q+1)$-invariant differential operators from ${\cal F}_λ$ to ${\cal F}_μ$ already known in the literature is obtained in a different manner.
We construct a natural conformally invariant one-form of weight $-2k$ on any $2k$-dimensional pseudo-Riemannian manifold which is closely related to the Pfaffian of the Weyl tensor. On oriented manifolds, we also construct natural conformally invariant one-forms of weight $-4k$ on any $4k$-dimensional pseudo-Riemannian manifold which are closely related to top degree Pontrjagin forms. The weight of these forms implies that they define functionals on the space of conformal Killing fields. On Riemannian manifolds, we show that this functional is trivial for the former form but not for the latter…
Local and global conformal invariants of submanifolds
Penn State University（宾夕法尼亚州立大学）；Rutgers University（罗格斯大学）；Wichita State University（威奇托州立大学）；University of Notre Dame（圣母大学）；Sun Yat-sen University（中山大学）
On a Möbius surface, as defined by D. Calderbank, we study a variant of the Einstein-Weyl (EW) equation which we call scalar-flat Möbius EW (sf-MEW). This is a conformally invariant, finite type, overdetermined system of semi-linear partial differential equations. We derive local algebraic constraints for this equation to admit a solution and give local obstructions. In the generic case when a certain invariant of the Möbius structure given by a symmetric tensor $M_{ab}$ is non-zero, the obstructions are given by resultants of 3 polynomial equations whose coefficients are conformal invariants …
Let $M$ be a smooth manifold equipped with a conformal structure, $E[w]$ the space of densities with the the conformal weight $w$ and $D_{w,w+\de}$ the space of differential operators from $E[w]$ to $E[w+δ]$. Conformal quantization $Q$ is a right inverse of the principle symbol map on $D_{w,w+δ}$ such that $Q$ is conformally invariant and exists for all $w$. This is known to exists for generic values of $δ$. We give explicit formulae for $Q$ for all $δ$ out of the set of critical weights. We provide a simple description of this set and conjecture its minimality.
We are interested in the study of the space of $n$-ary differential operators denoted by $\mathfrak{D}_{\underlineł,μ}$ where $\underlineł=(ł_{1},...,ł_{n})$ acting on weighted densities from $\frak F_{ł_1}\otimes\frak F_{ł_2}\otimes...\otimes\frak F_{ł_n}$ to $\frak F_μ$ as a module over the orthosymplectic superalgebra $\mathfrak{osp}(1|2)$. As a consequence, we prove the existence and the uniqueness of a canonical conformally equivariant symbol map from $\mathfrak{D}_{\underlineλ,μ}^k$ to the corresponding space of symbols as well for the explicit expression of the associated quantization m…
In this paper, we give an optimal inequality relating the relative Yamabe invariant of a certain compactification of a conformally compact Poin-car{é}-Einstein manifold with the Yamabe invariant of its boundary at infinity. As an application, we obtain an elementary proof of the rigidity of the hyper-bolic space as the only conformally compact Poincar{é}-Einstein manifold with the round sphere as its conformal infinity.
We extend the sphere theorem of \cite{CGY03} to give a conformally invariant characterization of $(\mathbb{CP}^2, g_{FS})$. In particular, we introduce a conformal invariant $β(M^4,[g]) \geq 0$ defined on conformal four-manifolds satisfying a `positivity' condition; it follows from \cite{CGY03} that if $0 \leq β(M^4,[g]) < 4$, then $M^4$ is diffeomorphic to $S^4$. Our main result of this paper is a `gap' result showing that if $b_2^{+}(M^4) > 0$ and $4 \leq β(M^4,[g]) < 4(1 + ε)$ for $ε> 0$ small enough, then $M^4$ is diffeomorphic to $\mathbb{CP}^2$. The Ricci flow is used in a crucial way to…
In this paper, we define conservative semibasic vector $1-$forms on the tangent bundle of a Finsler manifold. Using these vector $1-$forms, we characterize conservative $L-$Ehresmann connections with respect to the energy function. Then we find a correspondence between torsion-free semibasic vector $1-$forms and the subset of vertical vector fields. Taking into account this correspondence, we construct a class of semisprays that generates the Ehresmann connections mentioned above.
We derive a relationship between the eigenvalues of the Weyl-Schouten tensor of a conformal representative of the conformal infinity of a hyperbolic Poincaré manifold and the principal curvatures on the level sets of its uniquely associated defining function with calculations based on [9] [10]. This relationship generalizes the result for hypersurfaces in ${\H}^{n+1}$ and their connection to the conformal geometry of ${\SS}^n$ as exhibited in [7] and gives a correspondence between Weingarten hypersurfaces in hyperbolic Poincaré manifolds and conformally invariant equations on the conformal inf…
On a 3-manifold bounding a compact 4-manifold, let a conformal structure be induced from a complete Einstein metric which conformally compactifies to a Kähler metric. Formulas are derived for the eta invariant of this conformal structure under additional assumptions. One such assumption is that the Kähler metric admits a special Kähler-Ricci potential in the sense defined by Derdzinski and Maschler. Another is that the Kähler metric is part of an ambitoric structure, in the sense defined by Apostolov, Calderbank and Gauduchon, as well as a toric one. The formulas are derived using the Duisterm…
Conformally equivariant quantization is a peculiar map between symbols of real weight $δ$ and differential operators acting on tensor densities, whose real weights are designed by $λ$ and $λ+δ$. The existence and uniqueness of such a map has been proved by Duval, Lecomte and Ovsienko for a generic weight $δ$. Later, Silhan has determined the critical values of $δ$ for which unique existence is lost, and conjectured that for those values of $δ$ existence is lost for a generic weight $λ$. We fully determine the cases of existence and uniqueness of the conformally equivariant quantization in term…
Let $(M,g)$ be a pseudo-Riemannian manifold. We propose a new approach for defining the conformal Schwarzian derivatives. These derivatives are 1-cocycles on the group of diffeomorphisms of $M$ related to the modules of linear differential operators. As operators, these derivatives do not depend on the rescaling of the metric $g.$ In particular, if the manifold $(M,g)$ is conformally flat, these derivatives vanish on the conformal group $\Og(p+1,q+1),$ where $\mathrm{dim} (M)=p+q.$ This work is a continuation of [1,4] where the Schwarzian derivative was defined on a manifold endowed with a pro…
This paper has been withdrawn by the authors, as it was combined with "Conformally invariant energies of knots I" (math/0409396) to be "Conformally invariant energies of knots" which has replaced the former.
In this paper we study the existence and compactness of positive solutions to a family of conformally invariant equations on closed locally conformally flat manifolds. The family of conformally covariant operators $P_α$ were introduced via the scattering theory for Poincaré metrics associated with a conformal manifold $(M^n, [g])$. We prove that, on a closed and locally conformally flat manifold with Poincaré exponent less than $\frac {n-α}2$ for some $α\in [2, n)$, the set of positive smooth solutions to the equation $$ P_αu = u^\frac {n+α}{n-α} $$ is compact in the $C^\infty$ topology. There…
On locally conformally flat manifolds we describe a construction which maps generalised conformal Killing tensors to differential operators which may act on any conformally weighted tensor bundle; the operators in the range have the property that they are symmetries of any natural conformally invariant differential operator between such bundles. These are used to construct all symmetries of the conformally invariant powers of the Laplacian (often called the GJMS operators) on manifolds of dimension at least 3. In particular this yields all symmetries of the powers of the Laplacian $Δ^k$, $k\in…
In this paper we study the invariant Walker structures over the conformally flat four-dimensional homogeneous manifolds according to the Seger types of the Ricci operator.
We give a generalization of a theorem of Bôcher for the Laplace equation to a class of conformally invariant fully nonlinear degenerate elliptic equations. We also prove a Harnack inequality for locally Lipschitz viscosity solutions and a classification of continuous radially symmetric viscosity solutions.
In this note we study the problem of conformally flat structures bounding conformally flat structures and show that the eta invariants give obstructions. These lead us to the definition of an abelian group, the conformal cobordism group, which classifies the conformally flat structures according to whether they bound (higher dimensional) conformally flat structures in a conformally invariant way. The eta invariant gives rise to a homomorphism from this group to the circle group, which can be highly nontrivial. It remains an interesting question of how to compute this group.
The classical Rankin-Cohen brackets are bi-differential operators from $C^\infty(\mathbb R)\times C^\infty(\mathbb R)$ into $ C^\infty(\mathbb R)$. They are covariant for the (diagonal) action of ${\rm SL}(2,\mathbb R)$ through principal series representations. We construct generalizations of these operators, replacing $\mathbb R$ by $\mathbb R^n,$ the group ${\rm SL}(2,\mathbb R)$ by the group ${\rm SO}_0(1,n+1)$ viewed as the conformal group of $\mathbb R^n,$ and functions by differential forms.
The Yamabe problem concerns finding a conformal metric on a given closed Riemannian manifold so that it has constant scalar curvature. This paper concerns mainly a fully nonlinear version of the Yamabe problem and the corresponding Liouville type problem.
We show that the energy density of critical points of a class of conformally invariant variational problems with small energy on the unit 2-disk B_1 lies in the local Hardy space h^1(B_1). As a corollary we obtain a new proof of the energy convexity and uniqueness result for weakly harmonic maps with small energy on B_1.
This paper completes the construction of arbitrary order conformally invariant differential operators in higher spin spaces. Jan Slovák has classified all conformally invariant differential operators on locally conformally flat manifolds. We complete his results in higher spin theory by giving explicit expressions for arbitrary order conformally invariant differential operators, where by conformally invariant we mean equivariant with respect to the conformal group of $\mathbb{S}^m$ acting in Euclidean space $\mathbb{R}^m$. We name these the fermionic operators when the order is odd and bosonic…
In the article we introduce new conformal and smooth invariants on compact, oriented four-manifolds with boundary. In the first part, we show that "positivity" conditions on these invariants will impose topological restrictions on underlying manifolds with boundary, which generalizes the results on closed four-manifolds by M. Gursky and on conformally compact Einstein four-manifolds by S.-Y. A. Chang, J. Qing, and P. Yang. In the second part, we study Weyl functional on four-manifolds with boundary and establish several conformally invariant rigidity theorems. As applications, we prove some ri…
We establish blow-up profiles for any blowing-up sequence of solutions of general conformally invariant fully nonlinear elliptic equations on Euclidean domains. We prove that (i) the distance between blow-up points is bounded from below by a universal positive number, (ii) the solutions are very close to a single standard bubble in a universal positive distance around each blow-up point, and (iii) the heights of these bubbles are comparable by a universal factor. As an application of this result, we establish a quantitative Liouville theorem.
Conformally invariant functionals on the space of knots are introduced via extrinsic conformal geometry of the knot and integral geometry on the space of spheres. Our functionals are expressed in terms of a complex-valued 2-form which can be considered as the cross-ratio of a pair of infinitesimal segments of the knot. We show that our functionals detect the unknot as the total curvature does, and that their values explode if a knot degenerates to a singular knot with double points.
We completely resolve the boundary value problem for differential forms and conformally Einstein infinity in terms of the dual Hahn polynomials. Consequently, we produce explicit formulas for the Branson-Gover operators on Einstein manifolds and prove their representation as a product of second order operators. This leads to an explicit description of $Q$-curvature and gauge companion operators on differential forms.
Around 2007, A. Chang, J. Qing, and P. Yang proved a conformal gap theorem for Bach-flat metrics with round sphere as the model case. In this article, we extend this result to prove conformally invariant gap theorems for Bach-flat $4$-manifolds with $(\mathbb{CP}^2, g_{FS})$ and $(\mathbb{S}^2\times\mathbb{S}^2,g_{prod})$ as model cases. An iteration argument plays an important role in the case of $(\mathbb{CP}^2, g_{FS})$ and the convergence theory of Bach-flat metrics is of particular importance in the case of $(\mathbb{S}^2\times\mathbb{S}^2,g_{prod})$.
Let G be a finite connected simple graph. We define the moduli space of conformal structures on G. We propose a definition of conformally covariant operators on graphs, motivated by [25]. We provide examples of conformally covariant operators, which include the edge Laplacian and the adjacency matrix on graphs. In the case where such an operator has a nontrivial kernel, we construct conformal invariants, providing discrete counterparts of several results in [11,12] established for Riemannian manifolds. In particular, we show that the nodal sets and nodal domains of null eigenvectors are confor…
Conformal invariance plays a significant role in many areas of Physics, such as conformal field theory, renormalization theory, turbulence, general relativity. Naturally, it also plays an important role in geometry: theory of Riemannian surfaces, Weyl tensors, $Q$-curvature, Yang-Mills fields, etc... We shall be concerned with the study of conformal invariance in analysis. More precisely, we will focus on the study of nonlinear PDEs arising from conformally invariant two dimensional variational problems (e.g. harmonic maps, prescribed mean curvature surfaces, Willmore and Constrained conformal…
We study the conformally invariant variational problem for time-like curves in the $n$-dimensional Einstein universe defined by the conformal strain functional. We prove that the stationary curves are trapped into an Einsetin universe of dimension $2$, $3$ or $4$. We study the linearly-full stationary curves in a four-dimensional Einstein universe and we show that they can be integrated by quadratures in terms of elliptic functions, elliptic integrals and Jacobi's theta functions.
This work deals with the conformal transformations in six-dimensional spinorial formalism. Several conformally invariant equations are obtained and their geometrical interpretation are worked out. Finally, the integrability conditions for some of these equations are established. Moreover, in the course of the article, some useful identities involving the curvature of the spinorial connection are attained and a digression about harmonic forms and more general massless fields is made.
We define a conformally invariant action S on gauge connections on a closed pseudo-Riemannian manifold M of dimension 6. At leading order this is quadratic in the gauge connection. The Euler-Lagrange equations of S, with respect to variation of the gauge connection, provide a higher-order conformally invariant analogue of the (source-free) Yang-Mills equations. For any gauge connection A on M, we define S(A) by first defining a Lagrangian density associated to A. This is not conformally invariant but has a conformal transformation analogous to a Q-curvature. Integrating this density provides t…
We prove a classification theorem for conformal maps with respect to the control distance generated by a system of diagonal vector fields. It turns out that all such maps can be obtained as compositions of suitable dilations, inversions and isometries. We also classify all umbilical surfaces of the underlying metric.
On conformal manifolds of even dimension $n\geq 4$ we construct a family of new conformally invariant differential complexes. Each bundle in each of these complexes appears either in the de Rham complex or in its dual. Each of the new complexes is elliptic if the signature is Riemannian. We also construct gauge companion operators which complete the exterior derivative to a conformally invariant and (in the case of Riemannian signature) elliptically coercive system. These (operator,gauge) pairs are used to define finite dimensional conformally stable form subspaces which are are candidates for…
